= Point vortex in a wedge
For a <point vortex> of circulation $\Gamma$ in the wedge $0<\theta<\pi/n$, with positive integer $n$, the <conformal map> $z\mapsto z^n$ sends the domain to the upper half-plane. An opposite <image vortex at a plane wall>[image vortex] gives the <complex potential of a two-dimensional potential flow>[complex potential]
$$
w(z)=-\frac{i\Gamma}{2\pi}\log\frac{z^n-c^n}{z^n-\overline c^{,n}}.
$$
Remove the singular self-induced velocity before setting $z=c$. The resulting <complex velocity> is
$$
\frac{d\overline c}{dt}=\frac{i\Gamma}{2\pi}\left(-\frac{n-1}{2c}+\frac{nc^{n-1}}{c^n-\overline c^{,n}}\right).
$$
The finite first term follows from the expansion of $(z^n-c^n)/(z-c)$ at $c$. In polar coordinates this becomes $\dot r=n\Gamma\cot(n\theta)/(4\pi r)$ and $\dot\theta=-\Gamma/(4\pi r^2)$. Therefore $r\sin(n\theta)$ is constant along a <vortex> trajectory. The case $n=1$ is motion parallel to a plane wall, while $n=3$ describes a sixty-degree wedge.
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